Weather prediction is a complicated and complex process. Remote sensing instruments with progressively higher spectral and spatial resolution have been and are still being launched above the earth. These remote sensing instruments include the Atmospheric Infrared sounder (AIRS), the Tropospheric Emission Spectrometer (TES), the Infrared Atmospheric Sounding Interferometer (IASI), and the Geosynchronous Imaging Fourier Transform Spectrometer (GIFTS), etc. These instruments (or sensors, or sounders) provide thousands of channels for receiving spectral radiation from the atmosphere. These instruments provide a wealth of information on atmospheric properties and earth surface properties. In fact, so much information is provided by these instruments that it has become a challenge to analyze the vast amounts of received data.
For improving the accuracy of numerical weather prediction, accurate global observations of atmospheric temperatures and moisture profiles are needed. The satellite and airborne hyper-spectral infrared sensors, mentioned above, have the capability needed to achieve accurate numerical weather prediction.
At least two factors are considered for obtaining accurate atmospheric temperatures and moisture profiles from satellite or airborne observations. One is the accuracy and precision of the sounding spectrometer instrument. Another is the accuracy of the retrieval algorithm and the numerical approach used to process the data.
Retrieval of atmospheric parameters, such as temperature and water vapor profiles, from satellite or airborne infrared sounder systems are based on an atmospheric radiative transfer equation (RTE). The RTE equation is as follows:Rυ(η)=ευ(η)Bυ(Ts)τυ(η,0,ps)−∫0psBυ(T(p))(∂τυ(η,0,p)/∂p)dp+(1−ευ(η))τυ(η,0,ps)∫ps0Bυ(T(p))(∂τυ(η,ps,p)/∂p)dp+ρν(η,θ)τυ(η,0,ps)τν(θ,ps,0)Fυ cos θ  (1)
where                Rυ is the radiance observed from an instrument at wavenumber υ (cm−1);        ευ represents the earth surface emissivity at υ;        Bυ(T) is the Planck function at absolute temperature T (in Kelvins);        Ts is the earth surface skin temperature;        Tυ(η,p′,p) describes transmittance along the satellite view angle, η, of the atmosphere between a pressure level p′ and a pressure level p;        ps indicates the earth surface pressure;        Fυ is the solar irradiance; and        ρυ(η,θ) and τυ(θ,ps,0) are the solar bi-directional surface reflectance and the transmittance of the atmosphere along the solar zenith angle, θ, respectively.        
The atmospheric monochromatic transmittance τυ(η,p′,p) is defined as
                                                        τ              υ                        ⁡                          (                              η                ,                                  p                  ′                                ,                p                            )                                =                      exp            (                                                            -                  1                                /                g                            ⁢                                                ∫                                      p                    ′                                    p                                ⁢                                                      (                                                                  ∑                        i                                            ⁢                                                                                          ⁢                                                                                                    k                            i                                                    ⁡                                                      (                                                          p                              ,                              T                                                        )                                                                          ⁢                                                                              q                            i                                                    ⁡                                                      (                            p                            )                                                                                                                )                                    ⁢                                      sec                    ⁡                                          (                      η                      )                                                        ⁢                                                                          ⁢                                      ⅆ                    p                                                                        )                          ,                            (        2        )            
where                ki(p,T) is the absorption coefficient for absorber type i with an absorber mixing ratio qi;        g is gravity;        ki varies with temperature and pressure; and        the atmospheric absorber i may be water vapor, ozone, carbon dioxide, etc.        
It will be appreciated that a relationship between the radiance observed from a satellite or an airborne platform and a corresponding earth atmospheric temperature profile, or a corresponding atmospheric absorber profile may be established from equations (1) and (2). Given the atmospheric temperature and absorber mixing ratio at every pressure level p, including the surface temperature and emissivity/reflectivity properties, the monochromatic radiance may be calculated based on equations (1) and (2).
Equations (1) and (2) are strictly valid for monochromatic radiance for which Beer's law holds. Equation (1) is commonly used to interpret radiance observations by defining a spectral channel atmospheric transmittance function, which provides close agreement between the calculation and the observation. Monochromatic radiance, however, cannot be directly observed with a practical instrument. This is because a practical instrument has a radiance response that is not monochromatic and, instead, the instrument has a finite spectral resolution, even though atmospheric species emit (or absorb) radiance monochromatically. Most monochromatic absorption lines in the infrared region are caused by molecular vibration energy level transitions. These monochromatic lines are broadened in the atmosphere by molecular collisions, where the number of collisions depend on the atmospheric temperature and pressure.
The monochromatic RTE given by equation (1) provides an accurate model to determine the relationship between the radiance observed from a satellite or an airborne platform at a monochromatic frequency (wavenumber) and a temperature/pressure profile of the atmosphere. However, any instrument observed radiance has a finite spectral resolution such that the observed radiance is channel radiance rather than monochromatic radiance. Channel radiances may be determined by performing a spectral convolution of the atmospheric monochromatic radiance with an instrument line shape (ILS), or spectral response function, φ, as follows:
                                          R            c                    ⁡                      (                          v              ′                        )                          =                                            ∫                              Δ                ⁢                                                                  ⁢                v                                      ⁢                                          φ                ⁡                                  (                  v                  )                                            ⁢                                                R                  mono                                ⁡                                  (                  v                  )                                            ⁢                                                          ⁢                              ⅆ                v                                                                        ∫                              Δ                ⁢                                                                  ⁢                v                                      ⁢                                          φ                ⁡                                  (                  v                  )                                            ⁢                              ⅆ                v                                                                        (        3        )            
where                ν′ is the central wavenumber of the channel radiance.        For small Δν the following relationship holds:        
                                          τ            c                    ⁡                      (                          v              ′                        )                          ∼                                            ∫                              Δ                ⁢                                                                  ⁢                v                                      ⁢                                          φ                ⁡                                  (                  v                  )                                            ⁢                                                τ                  mono                                ⁡                                  (                  v                  )                                            ⁢                                                          ⁢                              ⅆ                v                                                                        ∫                              Δ                ⁢                                                                  ⁢                v                                      ⁢                                          φ                ⁡                                  (                  v                  )                                            ⁢                              ⅆ                v                                                                        (        4        )            
Many different channel radiative transfer models (RTMs) have been developed based on the monochromatic RTM. A recent detailed summary of channel RTMs is provided by Xu Liu et al., “Principal Component-Based Radiative Transfer Model for Hyperspectral Sensors: Theoretical Concept”, Applied Optics, Jan. 1, 2006, Vol. 45, No. 1, pps. 201-209. This summary is incorporated herein by reference.
Monochromatic RTMs use analytical formulae to simulate absorption line shape variation with temperature and pressure. More than 35 species with over 1,700,000 spectral lines have been measured for applications to the earth's atmosphere. Simulation of all monochromatic lines is very time consuming especially because one must account for all the different absorption lines that can affect any given frequency as a result of pressure broadening.
Spectral convolution of the monochromatic radiance spectrum using the instrument spectral response function produces an estimate of the observed radiance spectrum. This is called the forward problem, and it is well defined. Retrieval of atmospheric parameters from the observed radiance spectrum is called the inverse problem. The inverse problem is ill conditioned in the sense that many solutions may be obtained from one set of radiance observations. Statistical relationships between the atmospheric parameters and the spectral radiance measurements are commonly produced through radiative transfer simulation to provide a statistical constraint for obtaining an acceptable solution.